Well.....I can't give definitive answers. But I can offer a few thoughts.

A 50 cc scooter is a small scooter. Many of them are a good deal bigger. 150, 250 and 350 cc are common.

But if we imagine that we're talking about a 50 cc scooter.....then the combustion chambers are equal. Or near equal in the case of a 66 cc motor assisted bicycle.

I guess we could imagine one engine getting a more forceful power stroke by way of careful fuel/air mix. Or a better intake. But I don't suppose there's a whole lot of range to work with.

So if we imagine that the two power strokes in question are roughly equal, then there's only one way that one engine can possibly be more powerful than the other. That's by running faster. Higher RPMs, in other words.

So if a 50 cc scooter actually is more powerful than a 50 cc motor assisted bicycle, then faster engine speed is the only possibility that I can see. And it's believable. Picturing, say, a Yamaha 50 cc engine. If someone told me that it was built a bit better and can stand higher RPMs, I'd see no reason to doubt it.

But my (limited) experience with 50 cc scooters told me that they're not very powerful.

Honda Ruckus and Vespa in Ca. is a motorcycle, your main power source is a transmission.
According to my Ca. DMV, Motorcycle handbook, only a foot scooter with an engine is a "Motorized Scooter" with,
handlebars, platform for standing, human power, 2 wheels, and a motor, a seat but must not interfere with standing, helmet, CDL
and no DMV Reg.
If I read this right, no speed limits, no engine size limits? Transmission? I started building one last year from a 10" pit bike.
and yes you can use a bicycle frame.